Painlev\'e \uppercase\expandafter{\romannumeral34\relax} and collisionless shock in the defocusing NLS equation with step-like initial data in the transition regions

We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with step-like initial data.

Using the nonlinear steepest descent method, we derive the long-time asymptotic expansion of the solution to the Cauchy problem in three distinct transition regions.

In the first two transition regions, the leading-order asymptotics are characterized by Painlevé \uppercase\expandafter{\romannumeral34\relax}-type formula, while in the third one is a collisionless shock region, the leading-order asymptotics is describedin terms of Riemann theta functions.

Our analysis is based on the Riemann-Hilbert formulation associated with the Cauchy problem of the defocusing NLS equation.